Althought it is not obvious yet, this is the Hamiltonian necessary for a collection of “free" quantum particles. To make the connection more apparent, consider a canonical transformation to the pair of complex variables,This is also called a “Madelung transformation". See e.g., [1]}.
\(\Psi _{k} = \rho ^{1/2}\exp ik\Phi /\eta \quad \text{and}\quad \Psi _{k}^{*} = \rho ^{1/2}\exp -ik\Phi /\eta ~.\) 
